Finding exact solutions

Question

Find the exact solutions: ((3)/(8x+1))+((8)/(2x+5))=1

Answer 1

This amounts to a quadratic, as follow:

The common denominator is (8x + 1)(2x + 5) = 16x² + 42x + 5

6x + 15 + 64x + 8 = 16x² + 42x + 15 (and this is legitimate as long as neither of the factors in the common denominator are 0!)

Collect terms and divide by 2:

8x² - 14x -18 = 0

The roots (quadratic formula) are 9/4 and -1/2.